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Alternate coordinate systems

### Find the value of x or y so that the line passing through the given points has the given slope. (9, 3), (-6, 7y), $$m = 3$$

Alternate coordinate systems

### To plot: Thepoints which has polar coordinate $$\displaystyle{\left({2},\frac{{{7}\pi}}{{4}}\right)}$$ also two alternaitve sets for the same.

Alternate coordinate systems

### The system of equation $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{b}\right\rbrace}{\left\lbrace{e}\right\rbrace}{\left\lbrace{g}\right\rbrace}\in{\left\lbrace\le{f}{t}{\left\lbrace{\left\lbrace{c}\right\rbrace}{\left\lbrace{a}\right\rbrace}{\left\lbrace{s}\right\rbrace}{\left\lbrace{e}\right\rbrace}{\left\lbrace{s}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right\rbrace}{\frac{{{n}}}{{{\left\lbrace{2}\right\rbrace}}}}\backslash-{\frac{{{\left\lbrace{\left\lbrace{n}\right\rbrace}+{\left\lbrace{1}\right\rbrace}\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}{\left\lbrace{e}\right\rbrace}{\left\lbrace{n}\right\rbrace}{\left\lbrace{d}\right\rbrace}{\left\lbrace\le{f}{t}{\left\lbrace{\left\lbrace{c}\right\rbrace}{\left\lbrace{a}\right\rbrace}{\left\lbrace{s}\right\rbrace}{\left\lbrace{e}\right\rbrace}{\left\lbrace{s}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}\right\rbrace}$$ by graphing method and if the system has no solution then the solution is inconsistent. Given: The linear equations is $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{2}\right\rbrace}{\left\lbrace{x}\right\rbrace}+{\left\lbrace{y}\right\rbrace}={\left\lbrace{1}\right\rbrace}{\left\lbrace\quad\text{and}\quad\right\rbrace}{\left\lbrace{4}\right\rbrace}{\left\lbrace{x}\right\rbrace}+{\left\lbrace{2}\right\rbrace}{\left\lbrace{y}\right\rbrace}={\left\lbrace{3}\right\rbrace}.$$

Alternate coordinate systems

### What are polar coordinates? What equations relate polar coordi-nates to Cartesian coordinates? Why might you want to change from one coordinate system to the other?

Alternate coordinate systems

### The equivalent polar equation for the given rectangular - coordinate equation. $$\displaystyle{y}=\ -{3}$$

Alternate coordinate systems

### The reason ehy the point $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{\left(-{\left\lbrace{1}\right\rbrace},{\frac{{{\left\lbrace{\left\lbrace{3}\right\rbrace}\pi\right\rbrace}}}{{{\left\lbrace{2}\right\rbrace}}}}\right)}\right\rbrace}$$ lies on the polar graph $$\displaystyle{d}{i}{s}{p}{l}{a}{y}{s}{t}{y}\le{\left\lbrace{r}\right\rbrace}={\left\lbrace{1}\right\rbrace}+{\cos{{\left\lbrace\theta\right\rbrace}}}$$ even though it does not satisfy the equation.

Alternate coordinate systems

### Determine if $$\displaystyle\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash-{4}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{3}\backslash{5}\backslash-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace}$$ is a basis for $$\displaystyle{C}{o}{l}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash-{3}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{3}\backslash{7}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$

Alternate coordinate systems

### Consider the bases $$\displaystyle{B}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}\backslash{3}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash{5}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\right)}{o}{f}{R}^{{2}}{\quad\text{and}\quad}{C}={\left({b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\right)}{o}{f}{R}^{{3}}$$. and the linear maps PSKS \in L (R^2, R^3) and T \in L(R^3, R^2) given given (with respect to the standard bases) by $$\displaystyle{\left[{S}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{2}&-{1}\backslash{5}&-{3}\backslash-{3}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{\quad\text{and}\quad}{\left[{T}\right]}_{{{E},{E}}}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&-{1}&{1}\backslash{1}&{1}&-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ Find each of the following coordinate representations. $$\displaystyle{\left({a}\right)}{\left[{S}\right]}_{{{B},{E}}}$$

Alternate coordinate systems

### All bases considered in these are assumed to be ordered bases. In Exercise, compute coordinate vector v with respect to the giving basis S for V. V is $$\displaystyle{M}_{{22}},{S}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&-{1}\backslash{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{1}\backslash{1}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash{0}&-{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash-{1}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},{v}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{3}\backslash-{2}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}.$$

Alternate coordinate systems

### The intercepts on the coordinate axes of the straight line with the given equation $$\displaystyle{4}{x}-{3}{y}={12}.$$

Alternate coordinate systems

### All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. V is $$\displaystyle{R}^{{2}},{S}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},{v}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{3}\backslash-{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\$$

Alternate coordinate systems

### Given V = R, and two bases: B and C and the coordinator [v]B a) Find the change of coordinates matrix $$\displaystyle{P}{B}\rightarrow{C}.$$ b) Find the coordinator [v]C.

Alternate coordinate systems

### To find: The alternate solution to the exercise with the help of Lagrange Multiplier. $$\displaystyle{x}+{2}{y}+{3}{z}={6}$$

Alternate coordinate systems

### Let A and C be the following ordered bases of P3(t): $$\displaystyle{A}={\left({1},{1}+{t},{1}+{t}+{t}^{{2}},{1}+{t}+{t}^{{2}}+{t}^{{3}}\right)}$$ $$\displaystyle{C}={\left({1},-{t},{t}^{{2}}-{t}^{{3}}\right)}$$ Find tha change of coordinate matrix $$\displaystyle{I}_{{{C}{A}}}$$

Alternate coordinate systems

### All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. V is $$\displaystyle{M}_{{22}},{S}=\le{f}{t}{\left\lbrace{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{0}\backslash{1}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{1}\backslash{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace},{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{0}&{0}\backslash{0}&{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{r}{i}{g}{h}{t}\right\rbrace},{v}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}\backslash-{1}&{2}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}.$$

Alternate coordinate systems

### 1. Show that sup $$\displaystyle{\left\lbrace{1}−\frac{{1}}{{n}}:{n}\in{N}\right\rbrace}={1}{\left\lbrace{1}−\frac{{1}}{{n}}:{n}\in{N}\right\rbrace}={\frac{{{1}}}{{{2}}}}$$. If $$\displaystyle{S}\:={\left\lbrace{\frac{{{1}}}{{{n}}}}−{\frac{{{1}}}{{{m}}}}:{n},{m}\in{N}\right\rbrace}{S}\:={\left\lbrace{\frac{{{1}}}{{{n}}}}−{\frac{{{1}}}{{{m}}}}:{n},{m}\in{N}\right\rbrace},{f}\in{d}\in{f}{S}{\quad\text{and}\quad}\supset{S}.{e}{n}{d}{\left\lbrace{t}{a}{b}\underline{{a}}{r}\right\rbrace}$$

Alternate coordinate systems

### List the members of the range of the function h: x — 5 - 2x? with domain D = {- 2, - 1, 0, 1}.

Alternate coordinate systems

### To determine: a) Whether the statement, " The point with Cartesian coordinates $$\displaystyle{\left[\begin{array}{cc} -{2}&\ {2}\end{array}\right]}$$ has polar coordinates $$\displaystyle{\left[{b}{f}{\left({2}\sqrt{{{2}}},\ {\frac{{{3}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},{\frac{{{11}\pi}}{{{4}}}}\right)}\ {\left({2}\sqrt{{{2}}},\ -{\frac{{{5}\pi}}{{{4}}}}\right)}\ {\quad\text{and}\quad}\ {\left(-{2}\sqrt{{2}},\ -{\frac{{\pi}}{{{4}}}}\right)}\right]}$$ " is true or false. b) Whether the statement, " the graphs of $$\displaystyle{\left[{r}{\cos{\theta}}={4}\ {\quad\text{and}\quad}\ {r}{\sin{\theta}}=\ -{2}\right]}$$ intersect exactly once " is true or false. c) Whether the statement, " the graphs of $$\displaystyle{\left[{r}={4}\ {\quad\text{and}\quad}\ \theta={\frac{{\pi}}{{{4}}}}\right]}$$ intersect exactly once ", is true or false. d) Whether the statement, " the point $$\displaystyle{\left[\begin{array}{cc} {3}&{\frac{{\pi}}{{{2}}}}\end{array}\right]}{l}{i}{e}{s}{o}{n}{t}{h}{e}{g}{r}{a}{p}{h}{o}{f}{\left[{r}={3}{\cos{\ }}{2}\ \theta\right]}$$ " is true or false. e) Whether the statement, " the graphs of $$\displaystyle{\left[{r}={2}{\sec{\theta}}\ {\quad\text{and}\quad}\ {r}={3}{\csc{\theta}}\right]}$$ are lines " is true or false.

Alternate coordinate systems